Matrices

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Matrices — Quick Facts

A matrix is a rectangular array of numbers arranged in $m$ rows and $n$ columns, written as $A_{m \times n}$. The element $a_{ij}$ lies in the $i$th row and $j$th column.

Essential Types:

• Row matrix: $[3, -1, 4]$ — single row, $1 \times n$
• Column matrix: $\begin{pmatrix} 2 \ -5 \ 1 \end{pmatrix}$ — single column, $m \times 1$
• Square matrix: $n \times n$ (same rows and columns), e.g. $\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$
• Null matrix: Every element is zero, $O = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}$
• Identity matrix: $I_n = \begin{pmatrix} 1 & 0 & \cdots \ 0 & 1 & \cdots \ \vdots & \vdots & \ddots \end{pmatrix}$, diagonal elements = 1
• Diagonal matrix: Non-zero only on principal diagonal; e.g. $\text{diag}(2, -1, 5)$
• Scalar matrix: $\text{diag}(k, k, k)$ — diagonal with equal entries $k$
• Symmetric matrix: $A = A^T$, i.e. $a_{ij} = a_{ji}$ for all $i, j$; e.g. $\begin{pmatrix} 2 & 3 \ 3 & 1 \end{pmatrix}$
• Skew-symmetric matrix: $A^T = -A$, so diagonal elements must be zero; e.g. $\begin{pmatrix} 0 & 2 \ -2 & 0 \end{pmatrix}$

⚡ NDA Exam Tip: Questions on matrix types and properties appear frequently. If asked to identify whether a matrix is symmetric or skew-symmetric, always check both the shape (must be square) and the element-wise equality $a_{ij} = \pm a_{ji}$.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding and problem-solving fluency.

Matrix Operations

1. Addition and Subtraction Two matrices $A$ and $B$ can be added only if they have the same order ($m \times n$). Addition is element-wise: $$A + B = \begin{pmatrix} a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22} \end{pmatrix}$$ Matrix addition is commutative: $A + B = B + A$. It is also associative: $(A + B) + C = A + (B + C)$.

2. Scalar Multiplication Multiply every element by the scalar $k$: $$kA = \begin{pmatrix} ka_{11} & ka_{12} \ ka_{21} & ka_{22} \end{pmatrix}$$

3. Matrix Multiplication (Row × Column) The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. If $A_{m \times p}$ and $B_{p \times n}$, then $AB$ is $m \times n$. The $(i,j)$ entry of $AB$ is the dot product of the $i$th row of $A$ with the $j$th column of $B$: $$(AB){ij} = \sum{k=1}^{p} a_{ik} b_{kj}$$

Critical property: Matrix multiplication is NOT commutative in general. That is, $AB \neq BA$.

• $AB$ may be defined while $BA$ is not (dimension mismatch).
• Even when both exist, $AB \neq BA$ typically.

Example: Let $A = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$. $$AB = \begin{pmatrix} 1(0)+2(1) & 1(1)+2(0) \ 0(0)+1(1) & 0(1)+1(0) \end{pmatrix} = \begin{pmatrix} 2 & 1 \ 1 & 0 \end{pmatrix}$$ $$BA = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \ 1 & 2 \end{pmatrix} \neq AB$$

Associative law: $(AB)C = A(BC)$ Distributive law: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$

4. Transpose The transpose $A^T$ swaps rows and columns: $(A^T){ij} = a{ji}$. $$(A + B)^T = A^T + B^T$$ $$(AB)^T = B^T A^T \quad \text{(note the reversal!)}$$

Key Identity Properties: $$(A^T)^T = A, \quad (kA)^T = kA^T, \quad (AB)^T = B^T A^T$$

Determinant of a 2×2 Matrix: For $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, the determinant is: $$|A| = \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc$$

NDA Worked Example (2019): Find the determinant of $A = \begin{pmatrix} 3 & -1 \ 2 & 4 \end{pmatrix}$. $$|A| = (3)(4) - (-1)(2) = 12 + 2 = 14$$

Determinant of a 3×3 Matrix (expansion along first row): $$|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$$

Example: Find $|B|$ where $B = \begin{pmatrix} 1 & 2 & -1 \ 3 & 0 & 1 \ -2 & 1 & 4 \end{pmatrix}$ $$|B| = 1(0 \cdot 4 - 1 \cdot 1) - 2(3 \cdot 4 - 1 \cdot (-2)) + (-1)(3 \cdot 1 - 0 \cdot (-2))$$ $$= 1(-1) - 2(12 + 2) + (-1)(3) = -1 - 28 - 3 = -32$$

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory, derivations, and advanced problem types for thorough NDA preparation.

Singular and Non-Singular Matrices

A matrix $A$ is singular if $|A| = 0$. It is non-singular (or invertible) if $|A| \neq 0$. Only non-singular matrices possess a multiplicative inverse.

Adjoint and Inverse

The cofactor $C_{ij}$ of element $a_{ij}$ is: $$C_{ij} = (-1)^{i+j} M_{ij}$$ where $M_{ij}$ is the minor (determinant of the submatrix obtained by deleting row $i$ and column $j$).

The cofactor matrix is $[C_{ij}]$. Its transpose is the adjoint: $$\text{adj}(A) = [C_{ij}]^T$$

The inverse of $A$ is: $$A^{-1} = \frac{\text{adj}(A)}{|A|}, \quad |A| \neq 0$$

Verification: $A \cdot A^{-1} = I_n = A^{-1} \cdot A$

Example: Find $A^{-1}$ for $A = \begin{pmatrix} 2 & 1 \ 3 & 2 \end{pmatrix}$. $$|A| = (2)(2) - (1)(3) = 4 - 3 = 1 \neq 0 \quad \Rightarrow \text{ non-singular}$$ $C_{11} = 2,\ C_{12} = -3,\ C_{21} = -1,\ C_{32} = 2$ Cofactor matrix: $\begin{pmatrix} 2 & -3 \ -1 & 2 \end{pmatrix}$ $\text{adj}(A) = \begin{pmatrix} 2 & -1 \ -3 & 2 \end{pmatrix}$ $$A^{-1} = \frac{1}{1}\begin{pmatrix} 2 & -1 \ -3 & 2 \end{pmatrix} = \begin{pmatrix} 2 & -1 \ -3 & 2 \end{pmatrix}$$

Solving Simultaneous Equations via Matrix Inversion

For the system: $$a_1 x + b_1 y = c_1$$ $$a_2 x + b_2 y = c_2$$

Write in matrix form: $AX = B$, where $A = \begin{pmatrix} a_1 & b_1 \ a_2 & b_2 \end{pmatrix}$, $X = \begin{pmatrix} x \ y \end{pmatrix}$, $B = \begin{pmatrix} c_1 \ c_2 \end{pmatrix}$. Since $|A| \neq 0$, $X = A^{-1} B$.

Cramer’s Rule (special case of matrix inversion):

$$x = \frac{|A_x|}{|A|}, \quad y = \frac{|A_y|}{|A|}$$ where $A_x$ replaces the first column of $A$ with $B$, and $A_y$ replaces the second column.

NDA Worked Example (Cramer’s Rule): Solve: $2x + 3y = 8$ and $x - 2y = -3$. $$|A| = \begin{vmatrix} 2 & 3 \ 1 & -2 \end{vmatrix} = 2(-2) - 3(1) = -4 - 3 = -7$$ $$|A_x| = \begin{vmatrix} 8 & 3 \ -3 & -2 \end{vmatrix} = 8(-2) - 3(-3) = -16 + 9 = -7$$ $$|A_y| = \begin{vmatrix} 2 & 8 \ 1 & -3 \end{vmatrix} = 2(-3) - 8(1) = -6 - 8 = -14$$ $$x = \frac{-7}{-7} = 1, \quad y = \frac{-14}{-7} = 2$$

Rank of a Matrix

The rank $r(A)$ is the maximum number of linearly independent rows (or columns). It equals the order of the largest non-zero minor.

Key results:

• $r(A) \leq \min(m, n)$ for an $m \times n$ matrix
• $r(A^T) = r(A)$
• $r(AB) \geq r(A) + r(B) - n$ (Sylvester’s inequality)
• If $r(A) = n$ (full column rank), $A$ has a left-inverse
• If $r(A) = m$ (full row rank), $A$ has a right-inverse

Finding rank by elementary operations: Use row reduction (Gaussian elimination) to get an upper triangular form. The number of non-zero rows = rank.

NDA Previous Year Pattern: Questions on rank typically ask: “Find the rank of $\begin{pmatrix} 1 & 2 & 3 \ 2 & 4 & 6 \ 3 & 6 & 9 \end{pmatrix}$.” Here row 2 = 2(row 1) and row 3 = 3(row 1), so rank = 1.

Topic	NDA Weight (1–5)	Frequency
Matrix types & transpose	★★★★	Every year
Matrix multiplication	★★★★	Every year
Determinant calculation	★★★★★	Every year
Adjoint & inverse	★★★★	Alternating
Cramer’s rule	★★★★	Alternating
Rank of matrix	★★★	Occasional

Extended formula sheet: $$|kA_{n \times n}| = k^n |A|$$ $$|AB| = |A||B|$$ $$|A^{-1}| = \frac{1}{|A|}$$ $$(A^{-1})^{-1} = A$$ $$(AB)^{-1} = B^{-1}A^{-1}$$

The NDA Mathematics paper typically has 120 questions total; matrices and determinants together contribute approximately 8–12 questions, making them high-priority topics for any preparation strategy.